Fluctuations

13 Juin 2019 by René Lebon

“Our scientific hopes
have led each of us
to opposite extremes.

You believe in a God who plays dice while I believe solely
in the value of laws in a Universe where something
exists objectively, which I seek
to grasp in a wildly speculative manner.”

Einstein – Extract from a letter to Max Born in 1944.

“Fluctuat nec mergitur”
Motto of the city of Paris

Einstein and fluctuations

(Fluctuations: random, incessant spontaneous changes in physical systems in the vicinity of their state of equilibrium)

The media, who always twist things, took this extract from a letter and translated it as “God does not play dice”, and yet Einstein was far from being so abrupt. He balanced probabilities, which are fundamental in sub-atomic physics, and the apparent certainty of laws. Einstein thought that everything may be explained in terms of laws and that the probabilities applied to the physical world are not sufficient for the human spirit, which rejects chance. Reason must be able to triumph. It is a perfectly deterministic world, that of Spinoza, which Einstein fully adhered to. Everything has a cause, according to Leibniz’s principle of sufficient reason. This is Laplace’s thesis, according to which all events must be able to overlap each other in space and time, as part of a causal chain. You could thereby conceive of an ordered world, generated according to an original principle. Einstein was captivated by this idea that science must be able to explain everything. He spent the last years of his life attempting, in vain, to unify electro-magnetism and gravitation in order to better gain access to this ultimate explanation of the fields of nature. His determination to steer towards what seemed to him to be the truth is understandable.

From this, it absolutely cannot be deduced that Einstein was against introducing probability calculations into physical science. Around 1902-1904, he read “Lessons on the Theory of Gases” by Ludwig Boltzmann with great interest. Boltzmann described the state of a gas using entropy S, without knowing the states (velocity and position) of each of the molecules in this gas. Thermal equilibrium (temperature and pressure) is obtained by the most probable distribution of these states. Gas molecules in an enclosed space have a configuration (positions and velocities) distributed around a mean which firms up the temperature. We associate a probability (w) with each possible configuration of the molecules. Entropy S is proportional to the logarithm of this probability or, in other words, S = k log W (k being a constant called “Boltzmann’s constant). The gas’s state of energy relates to a set of elementary complexions or distribution of molecules thought of as being individually marked or discernible in each interval or velocity cell of equal probability. Boltzmann calculated the probability by counting the permutations of the molecules in the same cell.

Einstein saw this as a “general principle of nature”. He simply sought to interpret the probability; not in a combinatory form which, in his view, was too arbitrary, but in a physical form, seeing possible states of the system in this probability over time. Indeed, Boltzmann must have made the hypothesis that all the configurations have the same probability of occurring. He also had to admit that all microscopic states are accessible (the ergodic hypothesis). In 1903, Einstein published an article entitled “A Theory of the Foundations of Thermodynamics”. He noticed that k expresses the thermal stability of a system through its energy fluctuations. He wanted to go further into the idea of random fluctuations around a mean. This idea of fluctuations could relate just as well to molecules as to thermal radiation enclosed in a finite space; the black body.

Brownian motion enabled verification of these fluctuations. Einstein’s article was published in 1905, in the review Annalen der Physik, and was entitled “On the Movement of Small Particles Suspended in Stationary Liquids Required by the Molecular-Kinetic Theory of Heat”. Brownian motion was achieved by a particle in a fluid subjected to collisions from molecules in the fluid. The velocities of these molecules are distributed around a mean velocity in a bell curve, the shape of which depends on the temperature. This thermal agitation within the fluid produces a fluctuating force on the particle. The cumulative force of all these collisions obstructs the movement of the particle when it has a mean velocity in relation to the fluid. The fluctuating force due to the collisions tends to increase the temperature of the particle while the second effect of dissipation tends to bring its mean velocity in relation to the fluid down to zero and therefore reduce the temperature of the particle. The two effects lead to an equilibrium where the temperature of the particle proportional to its kinetic energy is just equal to the temperature it is floating in. Einstein showed that when this equilibrium is established, there is a relation between the diffusion of the particle and the characteristics of the medium, such as its temperature and its viscosity.

As it is possible to obtain the mean movement of the suspended particles and the fluctuations around this value, it is possible to deduce from this the physical magnitudes, such as Boltzmann’s constant k, Avogadro’s number, or the mean free path of the molecules in-between two collisions. Einstein thereby demonstrated the existence of atoms by using probabilities such as the frequency of events or of states, and by using fluctuations in these frequencies around their mean value.

The basic argument of Einstein’s 1905 article on the photo-electric effect related to fluctuations in light. Planck’s law, as formulated today, associates a whole number of light quanta (photons) with any electro-magnetic field mode. It gives the mean value for this number for a radiation frequency at a given temperature. To obtain this, Planck used a law set out by Boltzmann that links the entropy of a system to the number of possible configurations of system S=k log W.

Einstein demonstrated that the new way of counting configurations relates to a light consisting of quanta or, in other words, particles, each of them carrying an energy that is proportional on the one hand to Planck’s constant and, on the other hand, to the frequency of light.

Radiation can be compared to a gas because it too exerts pressure on a wall. It behaves as if it were composed of material points, light quanta with an energy proportional to its frequency. Einstein wrote: “What must however be borne in mind is the optical observations concerning the means over time and not the instantaneous values and, in spite of the full confirmation provided by the experiment, we fully understand that the theory of light operating with continuous functions in relation to space leads to contradictions with the experiment when you apply it to the phenomena of the production and transformation of light. According to the hypothesis to be adopted here, when a ray of light spreads from a point, the energy is not distributed in ever-increasing spaces; instead it consists of an infinite number of energy quanta located at points in space which move without becoming dissociated, and which can only be absorbed and produced in one go”.

A black body is a sealed cavity that is heated and is in a state of thermal equilibrium between the radiation emitted and the radiation absorbed by the electrons in the medium, which are considered to be resonators. Planck had admitted that the forms of radiation energy exchanged with the resonating electrons were discontinuous, and proportional to the frequencies in accordance with E = n h. A form of energy that resonates in an atom at a certain level will only be able to resonate at another level with a quantum leap proportional to the frequency.

From a static dynamic perspective, thermal radiation behaves as if it consisted of material points; light quanta. It must therefore be admitted that the light’s energy is distributed discontinuously in space by these quanta. Planck quantified the energy exchanges between matter and radiation. Einstein went further by quantifying the radiation energy itself.

In 1907, Einstein suggested that Planck’s law could describe the state of thermodynamic equilibrium of vibrations in materials.

In 1909, returning to the problem of light, Einstein calculated the fluctuations in light energy found in a delineated region of space. He demonstrated that the fluctuations are obtained by adding two contributions that respectively relate to a wave interpretation and to a corpuscular interpretation of light. He thus highlighted the wave-corpuscle duality for the first time, based on the most in-depth results of quantum theory.

In 1917, Einstein wrote an article that contained the discovery of stimulated emission (the laser). What was involved once again was a remarkable application of the statistical methods of Maxwell and Boltzmann to the theory of light. He described the elementary emission and absorption processes that are associated with transitions between the atomic levels, the existence of which had just been postulated by Niels Bohr. The absorption processes can only exist where light quanta are present, although the emission processes can occur in their absence. There are in fact two types of emission processes; processes stimulated by the presence of light quanta, and spontaneous processes that can occur even in a vacuum. This led to the first fully satisfactory interpretation of Planck’s law, which now stems from the kinetic equilibrium between the various emission and absorption processes, or which even comes down to assessment of the various transitions the atom makes between its levels.

Einstein studied the pulse transfer balances (or the movement quantity) between the atom and light and not just the energy transfer balances. He then demonstrated that each basic admission and absorption process is accompanied by a precise variation in the atom’s pulse. The sole explanation for this is that the light quanta transport not only energy, but also a pulse directed depending on the direction of the light’s propagation. The laws of conservation therefore impose a variation in the energy and in the atom’s pulse when a photon is emitted or absorbed. It is in this article from 1917 that the light quantum later called a “photon” is found for the first time, along with all its particle attributes, namely its energy and its pulse.

When absorption occurs, a photon is absorbed by the atom, which passes from its lowest energy level to a higher energy state. The probability of such a process is proportional to the starting level of an electrons population and the number of incident photons. As part of the emission process, a photon is emitted and the atom passes from its highest energy level to a lower energy state. The probability of such a process is proportional to the starting level’s population and the processes stimulated, or in other words, those triggered by the presence of incident photons. It is moreover proportional to the number of incident photons. For spontaneous processes, even those occurring in a vacuum, the probability does not depend on this number.

At a very low temperature, the atoms of a perfect gas are grouped together in the same state. They sort of freeze and become one. Identical mass material particles cease interacting. This is a minimum energy state.

Bose, an Indian physicist, sent an article to Einstein in which, considering electro-magnetic radiation to be a photons gas, he referred back to Planck’s law, which gives the frequency distribution of the energy of the electro-magnetic radiation for a black body; an enclosed space in a state of thermal equilibrium with a small hole drilled in it. Einstein applied this method, which was studied in the case of a photons gas, to perfect gases featuring massless particles, associated with electro-magnetic waves, with the behaviour of a perfect gas being that of identical material particles. This was the starting point for quantum statistical physics. The phenomena resulting from the law of large numbers are visible to the naked eye.

In the case of a photons gas in a state of thermal equilibrium within an enclosed space, what is called a mode is the mean number of photons in a well-defined energy state. This brings into play the temperature and the energy of the mode and applies to bosons.

Einstein considered the simple case of a perfect gas contained in an enclosed space. The waves associated with the material may be found in various modes or well-defined states of energy. The minimum energy mode is the fundamental mode where the particles have a nil velocity. The other modes are said to be excited. There is a maximum value for the mean number of particles in excited modes. The atoms abruptly gather together in the minimum energy state, creating a high-density zone where the atoms are trapped as if they were in a pit. This is a phase transition.

Each wave has a corresponding particle and Bose used this to calculate the mean number of photons in a state of thermal equilibrium within an enclosed space. His result brought into play both the temperature and the energy of the mode. The particles that are in the minimum energy mode have a nil velocity. The other modes are said to be excited and the particles in them have a velocity and therefore a kinetic energy that are nil. These excited modes feature a limited number of particles.

The six parameters for a light quantum are: the movement quantity vector and position characterising a space involving phases with six dimensions. When the temperature falls to absolute 0°, a number of molecules continually increasing in density become established in the first quantum state, which is the fundamental energy state. What is involved is degeneracy of the gas. The particles’ statistical independence is lost. They are tiered at the atom’s various levels in limited numbers.

In this article from 1916, Einstein proposed a theory for the exchange of radiation and matter.

There are three exchange processes:

1 – Absorption. The atom shifts to a higher energy state.

2 – Stimulated emission. The atom stimulated by an incident photon shifts to a lower energy level by emitting a photon.

3 – Spontaneous emission. The atom spontaneously shifts to a lower energy level by emitting a photon. According to Einstein, this event has no specific cause. Only the probability of its occurrence may be attributed to it. This shows that Einstein fully admitted the possibility of pure chance.

Jacobus Van’t Hoff found an analogy between osmotic pressure and the pressure of a perfect gas.

The molecules diluted in a solvent behave like the molecules in a perfect gas.

For Einstein, what is true for solutions is also true for dilute suspensions. This makes it possible to determine Avogadro’s number.

In articles published from 1902 to 1904, he set out his version of statistical mechanics. His attention was drawn to the fluctuations in a system in a state of equilibrium. While the irregularities in large systems disappear on average due to the large number of molecules comprising them, the fluctuations do create net deviations in relation to the mean thermodynamic values in systems containing a lower number of particles (all the same, what we are referring to here are several thousand or even millions of particles). It is hardly possible to highlight these fluctuations in a system on a microscopic scale under normal conditions.

Einstein sought situations where you could detect them in order to obtain Boltzmann’s number K and Avogadro’s number N. The number of gas molecules n equals n moles x N.

What you get, according to the law of perfect gases, is PV = nkT = RT.

R is measured. Determining k can give you N. Einstein thought that the black body is a physical system that would reveal the fluctuations.

To illustrate these different statistics, the one for bosons and the ones for fermions, let’s imagine five boxes and five particles.

If the five particles are discernible, which is the case with fermions, which are mutually exclusive, then there are 5 x 4 x 3 x 2 x 1 = 120 ways of distributing them in the five boxes. An arrangement involving one particle per box is more likely than five particles in the same box.

If the five particles are indiscernible, which is the case with bosons, which photons form part of, then there are five possibilities of placing five particles in five boxes. It is the first possibility that is most likely. The particles do therefore have a tendency of sticking to each other in the same box due to a statistical effect. The discernible particles follow Pauli’s exclusion principle and are tiered across the different levels depending on their maximum number per level. The statistical effect is called the “Bose-Einstein effect” in the case of bosons and the “Fermi-Dirac effect” in the case of fermions. In the first case, they come together, and in the second case, they disperse.

The fluctuations around a mean are dualistic phenomena because what is involved are alternations. For a physicist, a fluctuation is a random deviation from a mean, a an instance of heterogeneousness in relation to a homogenous situation. For example, it is the heterogeneousness of air (as it turns out, the nitrogen and oxygen molecules comprising it) which makes the sky blue.

We can also see through this study that Einstein was in fact far from being opposed to probabilities and therefore, to chance. Quite to the contrary, he saw it as a general principle of nature. The problem of qualifying the probability-based laws of nature leads to doubts regarding the certainty of those laws. As objects are groupings of atoms, it can only be concurred that, given the probability reigning in the sub-atomic field, we cannot see why you could conclude there is certainty in macroscopic laws. You would simply have to admit that the fluctuations that undoubtedly occur at microscopic scales smoothen out on large scales. In light of the large number of them, in the long term, the fluctuations end up by compensating for each other and you thereby get the impression that everything flows uniformly. But this is never anything more than an illusion. We know that energy can only be measured using small quantities. Does the same apply to space and time? Up until now, we have not been able to ascertain this, but this is doubtless due to the fact that this can occur on scales close to those used by Planck. For the time being, our apparatuses do not enable us to observe at this level. The quantum vacuum is probably the hub for this fluctuating agitation: these zero-point fluctuations that are the outcome of Heisenberg’s indeterminacy relations. For combined magnitudes like energy and time (you cannot have one without the other; it is dualistic), you could not obtain a zero value for one without the value of the other being infinite. That means that the product of them or, in other words, the action, cannot be greater than h, Planck’s constant, which characterises these measurements.

It must therefore now be admitted that what could be considered the most certain of laws does have a foundation within it that fluctuates randomly, even if this is not detectable. Obviously, we cannot say what Einstein’s reaction would have been had we succeeded in getting him to admit that the laws are probabilistic, laws for which he had the greatest admiration, thinking that a God immanent to nature was the source of all that.

This is doubtless the reason why he vehemently criticised the new physics coming out of Copenhagen, as represented by Niels Bohr.

According to this theory, you have to accept that a particle may be both here and there or that a door may be both open and closed.

When two particles interact, their properties combine and remain correlated even after the action has ended. By measuring a physical magnitude of one of the combined or “entangled” particles, you will automatically know the corresponding physical magnitude of the other particle.

A pair of objects prepared in an “entangled” quantum state behave like a single system only if the two objects are very far apart from each other. Their behaviour is indifferent to the space and time separating them.

In 1935, Einstein published an article with Podolsky and Rosen (which has, since then, come to be called the EPR paradox), claiming that the quantum formalism that predicted very strong correlations between two entangled particles, regardless of the distance separating them, was incomplete.

If the light beam passes through a polariser, the photons that pass through the polariser are polarised in the direction of the polariser (+1) or in a perpendicular direction (-1). There are no other possible results. An incident photon emerges either following the +1 route or the -1 route. Passing through the polariser determines the photon’s state of polarisation.

For a pair of entangled photons emitted simultaneously to the left and to the right, passing into two polarisers oriented in the same direction, the results will be fully correlated. If you have a +1 result for one of the photons, you will be sure to obtain +1 for the other ones.

If, during polarisation, the photons are very far away from each other, it may be admitted that they have exchanged a signal at a speed greater than light speed.

The polarisation of each of the entangled photons is not determined before measuring.

For EPR, this experiment showed that quantum mechanics is incomplete. There are hidden parameters that are still unknown and which need to be discovered.

In 1964, John Bell demonstrated that for two entangled particles, the quantum predictions are incompatible with any model explicitly integrating additional parameters that would determine the initial polarisation of each of the two photons upon them being emitted. It is not possible for all of the possible orientations of the two polarisers to reproduce the values for the correlations predicted by quantum mechanics. This was demonstrated by the experiment conducted in 1982 by Alain Aspect and his assistants. You should therefore stop thinking of the world as being formed of separate sub-systems with physical properties defined locally, and give up on a local, realist vision of the world of the sort that Einstein defended.

Einstein would have been satisfied with a world without probabilities with laws providing certainty. Regardless, he fully understood that there could be no certainty. This episode in the paradox made it possible to properly verify via experimentation that probability calculations reigned in the field of atomic physics.

As we noted, Einstein was greatly concerned with the problem of small-scale fluctuations and involving small quantities. His work interests us in the sense that matter and radiation are fluctuating, or in other words they oscillate around a mean value. Einstein thereby implicitly recognised the concept of modern dualism that we want to develop. Either the fluctuations add together and that can lead to an abrupt transition, a catastrophe, or, due to their large numbers, fluctuations end up compensating for each other. In the latter case, the value of the magnitude measured gradually flattens out around a mean, becoming smoothed out and lifeless. The world in which we live is the result of fluctuations that are characteristic of existence.